chapter03_250818_CCCCCCCCCCCCCC165052.pdf

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1. 34

CRE PER: 3

Strain Gage
Circuitry, Transducers,
and Data Analysis

INTRODUCTION

‘The electrical resistance strain gage is the most versatile of many devices to measure
strains on the surfaces of machine components and structural members Because resis
tance change in a strain gage is usually very small, it cannot he measured accurately
with an ordinary ohmmeter. The Wheatstone bridge is widely used in practice: one oF
more of the four arms of the bridge are strain gages In this chapter, Ihe basic principles.
of the Wheatstone bridge and the potentiometer are described, Such effects us due to
lead wire resistance are discussed briefly. Some transducers, in which the strain gages
are used as the sensing elements, are considered. Strain gage data analyses are pre-
sented. Finally, the nonlinearity of Wheatstone bridge and the variations of gage factor
are discussed briefly, because they are very important when large strains are 10 be
measured by using strain gages

3.2. WHEATSTONE BRIDGE

56

The Wheatstone bridge [1] is a basic circuit employed to measure extremely
small resistance changes in a strain gage when itis subjected to a strain, Figure 3.1
shows constant-voltage Wheatstone bridge that is normally used to record strain
gage outputs in static and dynamic applications. Let us consider briell the principle
of operation of the Wheatsione bridge. For the circuit shown in Fig. 3.1, the voltage
drops across R, und A, denoted by Vand Vu, respectively are given by the equar
tions

2. 32 Wheatstone Bridge 57

FIGURE 3.1. Coascentsotiae
Vesta brides.

v

en RER

here Vis the applied voltage across the bridge,
‘The voltage output of the bridge E is represented by
RyRy Roky
(Ry RVR RU)
Ii clear that the output voltage of the bridge is zero (Le, the bridge is Balanced) when
the term RR, RR is zero or when
RyRy = RR, 32)
Equation (32) represents a very important relationship which indicates that any
change in the resistance in one arm of the bridge can be balanced hy adjusting the
resistance(s) in the other arms of the bridge. Figure 32 shows some of the typical bal-
ancing arrangements.

Vas = Vas en

‘i all Hl

v Y y

in » (a
FIGURE 32 pics meti ofUalacios the Weston hit:

3. 58 Chapter3 Strain Gage Circuitry, Transducers, and Data Analysis

Consider an initially balanced bridge, namely. R,R; = R;R., so that E = 0, and
then resistances R., R, Rand R, are changed by the amounts AR, AR, AR, and AR,
“The voltage output AE of the bridge can be determined using Eq. (3.1): that i,

CRI + AR JR, + ARS) = (Re + ARR, + Ry)
(Ry + AR, + Ra + SRR, = AR, + Ry VARY
Using Eq. (32), neglecting second-order terms (eg. AR, AR,) and relatively smaller
terms (cg, R AR)) in the denominator, it can be shown that

aE

RR, (SR, AR: AR, ARs
e eee 5
a
Let RR, =m;then Eq. (33) can be rewritten as
oy” (AR _ AR: añ AR
leerer) es

Equation (3.4) is the basic equation governing the strain measurement of a
Wheatstone bridge. Note that the second-order terms should be included when the
strains being measured are greater than 5%. The correction for nonlinearity of the
‘Wheatstone bridge will be discussed in Section 3.6, The sensitivity of the Wheatstone
bridge, $,, cam be defined as,

MEY m (in an an ah)
Te elt mp AR RR
Normally, fixed voltage is applied to the bridge, then the sensitivity ofthe bridge
‘depends on the number of active arms employed, the page factor G, and the resistance
ratio m. However, a voltage magnitude can be selected to increase circuit sensitivity
The upper limit of the voltage is determined by the power dissipated by the strain
age(s) used in the bridge. Once a particular gage type is selected, the gage factor can-
‘ot be varied o increase sensitivity Finally, the wo most important parameters are the
number of ative arms and the resistance ratio Quarter. half, and full-bridge arrange
ments ae obtained if one, two, and four active strain gage arms are employed, respec.
tively: these are discussed in some detail below.

65

Quarter Bridge

‘The arrangement is called u quarter bridge when only one active strain gage is used, as
shown in Fig, 3.3. In this figure, R, is the active strain gage, which undergoes the same
deformation as the structure, and R, is the dummy or inactive gage, which is identical
to the active gage but does not encounter any mechanical strains and is used for com-
pensating the temperature effect; the other two arms contain fixed resistors. Any resis-
tance change AR, in the gage will disturb the balance of the bridge. The change of
resistance in the active gage may be produced by mechanical loading and/or by tem-
perature changes if the strain gage material's thermal expansion coefficient differs
from that of the structure's material undergoing strain analyses. Ifthe bridge is initially
balanced, from Eq. (3.4) the out-of-balance voltage AE will be

4. 32 Wheatstone Bridge 50

ib 7 i +
> |
| | se
hl
v
o o
PGURE33 Quatre armopeman
pa AR, _ VeGym 52
ER, tm
+m)? aE
ia en

VG,

Equation (3.7) shows that it is only necessary to measure the out-of-balance voltage to
obtain the strain when the bridge excitation voltage and the gage factor are known.

As mentioned earlier, temperature change will cause gage-resistance change so
‘that apparent strain will be introduced if the thermal expansion coefficients of gage
and structure or machine element materials differ. The simplest and most commonly
‘employed method to eliminate this problem is to use a dummy gage in the Wheatstone
bridge, as shown in Fig. 3.3b or e. The dummy strain gage is of the same type as the
active gage but is bonded to an unstressed part of a structure or machine element, or
on a separate piece of the same material as the structure or machine element under
Stress analysis; the active and dummy gages are located in close proximity so that they
experience the same temperature change. It is easy to show that the temperature-
induced apparent strain will be canceled out, and the bridge out-of-balance voltage is
related only to the strain due to mechanical loads.

Using Eqs. (3.5) and (36), we can write the sensi
ter bridge as

y of the fixed-voltage quar-

ee
“Te my

Gi es

Sey

Sometimes a variable voltage, whose upper limit is determined by the power dissipated
by the strain gage(s) placed in the bridge, is applied to the bridge to increase the circuit

For a variable-vollage quarter bridge, the voltage is related to the power
dissipation by a single page in one arm of the bridge as follows:

RU + m) = (1 + MP RO

V= IR, +

5. 50 Chapter3 Strain Gage Circuit, Transducers, and Data Analysis

where J, is the current in the active gage and P, is the power dissipation of the gage,

respectively Therefore, Ey. (38) can he rewritich as
Sou = PRE 69)

It is obvious that the sensitivity of the bridge is affected by two factors: the gage selec
tion controlling GA," and the circuit efficiency, denoted by mi(1 +m). It is easy to
show that the circuit efficiency for arrangements in Fig, 3.34 and © isthe same and ean
be increased by selecting a high value for m (eg, 8% with = 4 and 90% with m= 9).
“The supply voltage remains within reasonable limits, while the circuit efficiency of the
‘bridge, shown in Fig. 33, is fixed and equal to 51%, since m = 1 due to the location of
the dummy gage on arm 2."Thus the arrangement in Fig, 3.3¢ for temperature compen-
sation is better than that in Fig, 3.3b because the circuit efficiency remains the samo as.
that in Fig, 33a

EXAMPLE 3.1

A stain age is bonded o a tensile specimen along the axial direction to measure train in that
direction. To eliminate the effect of temperature change, a compensating page (or dummy) is
‘mounted on a separate picos of the same material as the specimen and placed clase 10 i, as
shown in Fig. E31. Prove that che apparent stains due to temperature change 47 will he cam
‘eled out ita quarter bridge, also shown in Fi, 3.1, is use.

0 == ae
Ki
1
y
une a

Solution During the application ofa tensile loud P and temperature change AT, strain
will undergo a resistance change consisting of two pars, namely,

AR, _ Rs + Ayr

BR A,
‘whore AR, and AR, represent resistance change uf the active gage due to load P and tempera:
une change 47 respectively However, strain gage 2 will undergo a resistance change due solely
to temperature change Thus

BR, AR
RR

6. 32 Wheatstone Bridge 61

ata Ge
ticas A, = AR IR, if wo pagos are Hen (R, = and they underge dete)
temperature cage AT Ge. AR,, = SR) Ts
va
18

se

ich is independent of temperature change 4% 1
achieved by employing a half- or a full bridge circuit, which will be lus

EXAMPLE 32

‘Two strain gages are placed in a series in arm R, of a fixed-voltage quarter Wheatstone bridge,
shown in Tig, E32. Assume that two pages experience the same stains Determine the circuit
sensitivity and compare it with the value obtained if single gage is employed,

M
igure 2 Y

Solution ora fixed-woliage quarter Wheatstone bridge with a single active page, the ci
Sensitivity $.,is given by Eg. (38), namely,

ip em EA
Aer m

Er

vay

For a single strain gage, ARK, = ARR. where R, is the resistance of à strain gage Casually
120 Mor 350.1). For the case of two ges connected in a series,

AR, AR, + AR, AR,
ho RR m

Ths there is no increase in circuit sensitivity with 10 gages com
sensitivity, in cach ease is given by

ted in a series The circuit

mo var,
s, r=

ee

7. 62 Chapter3 Strain Gage Circuitry, Transducers, and Data Analysis

EXAMPLE 33

Suppose that the voltage can be varied in the Wheatstone bridge shown in Fig, E32. (a)
‘Compare the circuit sensitivity for the two gages connected in a series with the one for a single
ase. (b) Determine the percent change in circuit sensitivity for the two gages over a single gape.
(6) Determine the required voltage forthe two gages and fora single gage il P, = LU WR, =
1200 and m =7,

Solution (a) Fora variable-voltage quarter bridge, the sensitivity is given by Eg. (39), namely,

Secure
Tora single active age, =P, and Ry = Ras
se 2 =P and Ry= By
Shy = Poy PRE

For two gages in a series P,=2P_ and R, = 28, therefor,

ei
Tem

argon = 2 Jar" ~ 254,

Sotho ses for the 0 age ina series ice ht or a singe gage. lb seen shortly

thats sve to the u other vage two gages peri eun fella wc in mage

ta for aigle gags vo le it by power disipa nu gle gas
Change Si, 8 = mr.

OL PER) SGD) = 00158 As for single page, Ry =
Vi = AUR, + Re) = DR 0) = (00188) (120) {1 +7) = 158 ¥
and forthe to gages R, =R, +R
LL + m) = (00158) (120 + 120) 1 + 7) = 3036

Half Bridge

Ifthe dummy gage in Fig. 3.3b or e is replaced by an active gage, as shown in Fig. 3.4,
the resulting arrangement is called a half bridge. The hall bridge has advantages for
temperature compensation and higher bridge sensitivity over the quarter bridge so
that small strain levels can be detected more accurately. As with the quarter bridge, the
arrangement in Fig. 2:4b is superior to that in Fig. 34a, since the circuit efliciency in
Fig, 3.4b is usually higher ifa variable voltage is applied to the bridge.

Figure 3.5a shows an example of half-bridge application: A circular specimen is
subject to a tensile load. Gages a and ¢ are mounted along the axial and transverse
directions, respectively. The corresponding bridge connection is shown in Fig. 3.5b (the
half bridge). Assuming chat identical strain gages ure used, gage a has a resistance
increase of AR at some load, and gage twill have a resistance decrease of AR due to.
Poisson's effect, where j is Poissan’s ratio.

8. 32 Wheatstone Bridge 63

se
CURE 34 Haig ronge
Using Eg, (3) the out-oftalance voltage AE becomes
(ge) fm
‘Therefore, the sensitivity ofthe bridge i
NED APR) 6)

Tim

U + mp

Comparing Eq. (3.10) with Eqs. (38) and (3.9), we find that che circuit efficiency
increases (1 + u) times. In practice, the principal advantage of this type of arrangement
is to obtain automatic temperature compensation. Also, if two puirs of gages are
‘mounted in a diametrically symmetsic direction, then employing a half-bridge circuit
by connecting the corresponding gages in series, or using a full bridge (as in tens
compression load cell), we can automatically cancel any bending effect. It should be
mentioned that the self-temperature-compensating gages described earlier must he
used when the specimen is large or when temperatures vary greatly.

bl
® o
FIGURES. Specimen in mal oo wth Sige connection

9. 64 Chapter3 Strain Gage Circuitry, Transducers, and Data Analysis

EXAMPLE 3.4

‘An assembly ula steel bolt and an aluminum tube is subjected to a temperature change 47. To.
measure the mechanical strain in the tube due to temperature increase AT, four gages are
bonded in the longitulinal ana circumferential directions, as shown in Fig E3.da: gages € and D
are placed diametrically opposke to gapes A and respectively, half bridge with fixed voltage
Vi adopted How should the four ges inthe bridge be placed to eliminate the apparent steam
due to temperature inercase and to measure the mechanical strain? What is the output DE ia
terms of YG, y and €? (e isthe mechanical strain in the axial direction due 10 temperature
increase AT)

+

wo) 22

FIGURE ES

‚Solution "Ihe our gages are positioned in the Wheatstone bridge as shown in Fig, E Ab

Ra

Ry = Rem Ry = Ry

Since all gages experience the same temperature change, the resistance change AR, due 10 a
mismatch ofthe $-T-C number (thermal expansion coefficient) vi! be the same, Using Ey. (34),
these resistance changes due to temperature will he canceled, In other words the bridge is te
perature compensated

The mechanical strain in the axial direetion is et is ue in the circumferential direction due
10 Poisson's fet. Ths

AR, AR, + AR: MR,
Ri Rat Re OR, À
SR: _ Re ARp_28Re_ Re
ERE AA

Note that op = 1 and Akt, = AM, = O,and by using Eg (RAILAE = (VIGA + pe.

EXAMPLES.S
In Example 21 it was mentioned thatthe gage shown in Fig. E2. could alo give readings pro
‘portional fo shear tral y it was mounted along the avis of the shaft anıl connected in a half
bridge. asshown in Fig. AS. Verify this statement

10. 32 Wheatstone Bridge 65

FIGURE ERS

Solution Using Ka (134a).we have

es costar 2026
ul en

024

aR,
me Ge

Since Ry = Ra = Rom = 1. Note that AR, = AR =

and by Bg. (3.8), We obtain

Vr a
isin

Y (Aa _ Ae

OGG, ma

Full Bridge
‘The highest circuit efficiency can be achieved it four active gages are employed in the
bridge, as shown in Fig. 6. Such an arrangement is called a full bridge. Lis automat
cally temperature compensated when all four active gages bonded on the same mater-

FIGURE 3.6 Full vide rangement

11. 66

Chapter 3 Strain Gage Circuitry, Transducers, and Data Analysis

bl
o o
FIGURE37. Specinen intending wi abri rangement

I, ad are thus subjected tothe same temperature changes and are ofthe same type.
Im this case also, bridge balancing should he obtained by using the arrangements
shown in Fig. 320 or e. Use of a full bridge can increase the circuit efficiency up to
200% over that of quarter bridge. The following example ¡Nustrates this.

Figure 3.7 shows four identical train gages mounted on a beam with rectangular
ross section in pure bending and with full-bridge connections The strains atthe top
and bottom surfaces are the same in magnitude but opposite in sign, Therefore, ages a
and € will have a resistance increase of AR during bending, while gages b and d will
have a resistance decrease of AR. Since resistance changes due to. the same lem-
perature change will bo canceled out automatically by using Ey, (34) and noting that
m = 1,1 outot-balance voltage AF hecomes
(Une SAE q

AR o

AE

Therefore, the sensitivity ofthe bridge is
Sup = VG, = 2G(P,P,)'? em

Note that this is possible only for the bending case where two gages measure positive
strain and the other two determine an equal magnitude of negative strain.

If in addition to the pure bending moment, the structural member is subjected to
a tensile force, the four gages will experience the same amount of resistance change
due to this axial force, in addition to change caused by the bending moment. Using.
Eq. (34), it can be shown that the effect of the tensile load will be eliminated in the
full-bridge arrangement, leaving only the bending components to be measured,

EXAMPLE 3.6

A load cell is to be designed to measure a tensile force P tha i applied along the axis of the
structural member (load cell) as shown in Fig, 37a. The bending moment, shown inthis ligue.
may be produced due to light missligment of load with this aus. Tu measure the tensile strain
caused by load P alone, how should the four pages be placed in a Wheatstone bridge? Neglect
the effec of temperature chango.

12. se

3.2 Wheatstone Bridge 67

FIGURE ES

Solution Since the temperaturo-chango cost is negligible. a quarter bridge can be employed,
as shown in Fig. E36 (all four gages are connected in single arm). Let the resistance change
‘caused by pure beading be AR y and the resistance change caused by tensile load P be AR, Each
age has a resistance change AK, which isthe sum of AR and AR o

For gages a and €;

ak,

8e + Su
and tor gages and
AR = Ake Ale
Terres Ea),
Vi AR Vin (SR; + ARy) + (Ry — Ru) + (AR > + AR y) + (ARr — AR)
Ten R Gen RoR RAR
Yo SBR, Ve ARE. Um

2 Ge
dtm)! 48 + my Ry 4m

where R, G, and are the gage resistance, the gage factor, and the axial strain, respectively. and
ARR sopresents the 1oal resistance change por nit resistance in a single arm.

It should be pointed out here that there is no circuit sensitivity increase due to
the use of four gages in a quarter bridge. Therefore, itis not economical to use four
gages to measure strain in the axial direction, To eliminate any possible bending effect,
only two gages are recommended in practice, These two gages should be connected in
series and positioned in a quarter bridge. The best design for this load transducer to
maximize the circuit sensitivity is two gages in the axial direction and two in the trans-
verse direction with a full bridge.

EXAMPLE 37

Four strain gages were bonded on a cantilever beam and positioned in a fixed-voltage
‘Wheatstone bridge as shown in Fig. E37, Prove that the bridge output is proportional the
applied lending moment M atthe fixed end respecto ofthe point of application of load P as
long us it remains to the right of gage de

13. 68 Chapter 3 Strain Gage Circuitry, Transducers, and Data Analysis

7

x

igure 637
Solution From Fig. E27 it sun be seen that
LL, = PRL Ae
and
_ Px
>
here £ is Young's modulus and Ls the moment of inetia or the second moment of area},

respectively, Using Eqs (24) and (2.9) and noting that ne

AR, AR, AR ARs)
R&R Ni

a"

Gr)

2
LOL + A) (-2L - MOLA

ren

whore Aixa constant and is definition is apparent

3.3 CORRECTION FOR LONG LEAD WIRES

A Wheatstone bridge can be used to measure extremely small resistance changes in
strain gages Any small ch
will affect the output volta

14. 33. Correction for Long Lead Wires 68
Ry

La

ar

Ti
FIGURES. Two leadwire rangement Y
Consider first a single gage with two long lead wires a shown in Fig. 35, From
the figure its clear bal
Ri Ret 2K,

where R, and R, are gage and lead wire resistances, respectively The lengths of the
two lead wie ae assumed to be the same, which is the usual casein practice, Note
that

aR, AR RR, Ge

Ry RI, 14 IRE, 17 RR,
‘where Gi the gage factor and e and € arc the correct and indicated strains respec-
tively, Equation (3.12) clearly shows that error will be introduced in the strain read-
ing the attenuation i o be less than 0,5%. R,/R, should be les than 0.025. Thus
fora 12042 gage. Ry should be les than 0.30

“To compensate for the error introduced by the lead wires, the gage factor should
be set at G7 which is calculated by the following equation:

Ga)

G

TH 2R,

where Gs the gage factor given by the manufacturer.

R,

EXAMPLES.

A each lead in Fig 38 is a 28-N-dong copper wire with 40.05: n. diameter, what isthe lol resis
tance of the leads? Ita strain gage with a resistance of 120-0 and a gage factor of 2.16 is used,
‘what modified gage factor should be used ina strain indicacor in ordes to read directly the “eor-
rest” srain of the specimen? The copper has a resistance of OAI17-0 per meter of length for u
av conductor

Solution \ m~3281,1 mm? = DODISS in The cross-sectional area ofthe copper lead is given
by

003)"

15. 70 Chapter3 Strain Gage Circuitry, Transducers, and Data Analysis

‘Thos
_ Does 228)
"7 9,000707/0.00185
‘The total resistance ofthe leads is 0.568 0:

usa

‘Therefore, the gage factor should be se at 2:15 on the strain indicator

Note that if 350.0 strain gage is used, the error introduced by the leads will be
reduced. Using a larger-resstance strain gage wil also be helpful in initially halancing
the bridge.

Temperature compensation can be lost if long lead wires are used with Ihe von-
nection shown in Fig. 38, Suppose that gage à is the dummy gage used to compensate
the temperature effect. Assume that gage 1, gage 4, and the lead wires are subjected 10
temperature change AT when the strain is being measured by gage 1. The out-of-
balance voltage ean be expressed as

y 1 (AR Rias | 28Resr Ar
ee en
where R, = R, + 2R, is the resistance of arm 1 and Ry = R, is the resistance of the
dummy gage with negligible resistance ofthe lead wires These four terms represent the
resistance change inthe active gage due to the applied load, the resistance change in the
active gage due to temperature change 41 the resistance change in th long lead wires
due to temperature change AT, and the resistance change in the dummy gage due 10
temperature change AT. Its clear that temperature compensation cannot be achieved
since in general the sum ofthe second, (he third, and the fourth terms cannot be zero.

613

EXAMPLE 3.9

Suppuse that copper has a temperature coellicient ol resistivity of 00022 °F calculate the
apparent train component due to temperature change of Fin the Lead wires for Example 38.

Solution Due the 1°F temperature change, the total resistance change ofthe lead wires is
AR, = 0.568 x 0002 = 000125 0

Tus
AR, _ boot
RG, ana

= Ape per E

To achieve temperature compensation, three-lead-wire connection, as shown in
Fig 39, should be employed when the active gage and compensating gage (either
dummy gage or another active gage) can be placed adjacently. The three-lead-wire sys-

16. 34 Strain Gage Transducers 71

all
u

FIGURE 39 Three ia wire arrangement

tem may still have a problem of signal attenuation; however, this attenuation is
reduced by a factor of approximately 2 compared with the two-lead-wire system. Ifthe
adjusted wage factor G; is used to eliminate the error, it can be shown that G; is given
hy the equation

LE
STR
It should be pointed out that G;
sage or an active gage.

To show that temperature compensation can be achieved in a three-lead-wire
system, lets consider the case when gage 4 is a dummy gage. Equation (3.13) then

the same regardless of whether gage 4 is a dummy

becomes
mo {4 ARosr , 28Risr ias 2)

tn pele CAD

Its obvious that the sum ofthe terms related to temperature change is zero if R= R.

= R,+2R, and if the gages and lead wires ure subjected to exactly the same tempera:
ture change,

3.4. STRAIN GAGE TRANSDUCERS.

A strain gage transducer is a device that usos strain gages as the sensor to produce an
electrical signal that is directly proportional to such mechanical quantities as force, dis-
placement, pressure, torque, and acceleration. In principle, Examples 3.5, 3.6, and 3.7
are applications of strain gages for transducers. Many different types of sophisticated
strain gage transducers are commercially available. A few transducers—namely, load
cell, torque meter, accelerometer, and displacement transducer—ure shown in Fig,
10. The operating principles of a load cell and torque meter are described in this sec
ion.
Figure 3.10a shows a tension-compression load cel. To increase sensitivity and
accuracy (10 eliminate possible bending or torsional effects), four strain gages are
‘mounted on the central region of the bar with two gages in the axial direction and In0

17. 2 Chapter3

Strain Gage Circuitry, Transducers, and Data Analysis

ma 7)

‘Gages and Bon the oppaie ide

ih —

examples tai gage trnsducot (a Lesion compression oad el) extemomter;

18. 34 Strain Gage Transducers 73

‘gages in the transverse direction and connected in a full bridge; the two gages in each
sel are bonded at diametrically opposite locations When a load P is applied to the bar,
the axial and transverse strains are

P ye
“TEA FA

modulus of the bar, À the cross sectional arca of the member, and

us, 615

where Bis Young!
is Poisson's ratio.

arms R, and R, are gages mounted in the axial direction, and arms R; and Ry
are gages mounted in the transverse direction, the changes of resistance in four gages
can be determined by using Eq. (3.15):

AR,
Re

¿Reg POR
R
Substituting Eg. (3.16) into Eg. (34) and noting that m = 1 gives

6.16)

From the equation above, i is clear that the relation between out-of-balance voltage
and applied loud P is linear if the deformation of the bar is elastic, so it can he cali-
rated to read the load direct

To convert a cylindrical piece into a torque meter, four gages A, B, G and D are
‘mounted on the center portion of the piece with two gages (A and C) oriemed at an
angle 45° with respect to the axis of the shaft (the x axis) and two gages (B and D) ori-
tented at an angle -45° with respect to the axis of the shalt, as shown in Fig. 3.104.
Consider two strain gages À and B the strains along the gage axes are obtained by
using Eq. (1.34a) as

From the equations above, the shear strain +,, is

tw _ Te
nungen em
where Gis the shear modulus of the material of he structural member, 7 the applied
torque, the radius of the cylindrical member, and J the centroidal polar moment of
inertia ofthe cross section, respectively.

Let arms R, and R, be gages À and €; and arms R, and R, be gages B and D. The
changes of resistance in the four gages are

aR ARG AR AR
ROR RR

“he

€ am)

19. 74. Chapter3 Strain Gage Circuitry, Transducers, and Data Analysis

Substituting Eq.(3.17) and Eq, (3.18) into By. (3.4) and noting that = 1 gives
VGr
Der

I is obvious that the relation between out-of-balance voltage and applied torque Tis
linear ifthe deformation of the shaft is clastic. Note that the device is essentially inde-
pendent of the effects of axial and bending strains so itis often called a torque meer.

3.5. STRAIN GAGE ROSETTE DATA ANALYSIS AND CORRECTION

Strain gages are normally used to determine the plane stress states at à point on the
free surface of a structural or machine part subjected to applied loads, Let the x and y
‘axes be inthe plane of free surface and the 2 axis be perpendicular to it; en ur, =
+, = In general, a three-element rosette is required to determine the stresses 0,3,
aud +. since the directions of the principal stresses are normally not known, Thus
there dre three independent unknowns:¢,,.,, and, :altemalively, the unknowns are
the principal stresses a, and o, und the principal angk ses the directions
‘of the principal stresses are known beforehand, and then only a two-element rectangu-
lar rosette is needed. The rosette is bonded to Ihe surface so that each gage is lined up
‘with a principal direction,

Consider a general three-element rosette, whose orientations are shown in Fig
3.11. By using the equations of strain transformation [ie., Eg. (1.34). we obtain

4,608 By + € sin? + y, sin By cos By

+ 6,008 Bu + 6 sit Pa +, sin Ba 08 Br 649)
ean = 6.008 Bay Y ey sin? Buy +, Sin Bi COS But
For a given set ol angles By and Py. the strains €, €, and y, can be obtained by

solving Eqs (3.19) in terms of the measured strains e, €y, and €, Therefore, ho prin
pal strains e, and e, and the principal angle a, (hetween e, and the x axis) can be
obtained hy Egs (1.37) and (1.38), which are rewritten as

FIGURE 3.11 Orientation af thee
clement mie a

20. 35. Strain Gage Rosette Data Analysis and Correction 75

1
azi tots

+%
1 1 de
onze te) Flee? +m)" G20)
1 »
sie nt Xe e)
u = =

Once the principal strains « ‘Young's modulus E, and Poissor's ratio ju are
known, the principal stresses 0, and 6, can be found by replacing the subscripts x and y
in Eq.(1-47) by I and 2:namely,

sé + wed
2 62)

Note that the directions of the principal stresses are the same as those of the principal
strains

EXAMPLE 3.10 17]

By considering the equilibrium of the free-body diagram, shown in Fig E3.10, derive an expres

1 tu compute the principal angle win teem of principal stress and Cartesian stress compo-
nent at à point. Then prove that the principal angle ean he comvenicntly computed by the
following equation [the last equation in Eq (320):

FIGURE £3.10 1

21. 76 Chapter3 — Strain Gage Circuitry, Transducers, and Data Analysis

Solution Without loss of generality, we assume thal the dimension of the free boxy in the =
direction equals 1.The equations of equilibrium ofthe free body in the x andy directions are

3F,= a
ET
15,209 5
Since tana = ay, thus

By replacing 0, a,.0,, and, With, e, 2, and y, 2 in the above equation, we obtain
qe

E]

The two most commonly used three-element rosettes are the rectangular rosette and
the delta rosette. Consider first the three-element rectangular rosette, For mere conve-
nience, let the rosette be positioned at the angles of 0%, 45°, and 80", as shown in Fig
312

FIGURE 3.12 Gage orientations ofa
threoelment rectangular rset

with B,

Ba = 45* and y= 90°, Eg. (8.19) becomes
To

6

22. 35, Strain Gage Rosette Data Analysis and Correction 77
Solving the equations above shows that the shear strain is

Yo © Zen 4 = en 623)
) into Eg, (320), he principal strains and the princi-

Substituting Equ. (3.22) und (
pal angle can be written as

webride epee?

P + Dee

Stet eu) = q le ea ae 6»

0 = tant

EU
as = a, +

Substituting Eq. (3.24) into Eq, (321), the corresponding principal stresses can be
expressed in terms of the measured strains as follows:

5 ter = em)? + Qu = 65 = em]!

629
Les ui + Qu = eu}

‘EXAMPLE 3.11

A rectangular theee-element cosette was mounted on an aluminum structure The three sra
readings are € = 40g, cu 1% on ey, = 120, Find he values nd dicctions ofthe pria-
cipal strains and the values of the principal stresses Young’s modulus and Prison’ ratio are
D kei and 0.38, respectively

Solution
es bu = 1600
E en = One

Zen = ec = em = {2 X 1800) = 400 — 1200 = 2000p

05; = en)? + Gey = € = en) OSO] + (2000) = 1077ue

Using Eg. G24). we get

zn
= 1877 — 1200)

as = a) + 90 = 196%

23. 78 Chapter3 Strain Gage Circuitry, Transducers, and Data Analysis

‘The principal tresses, and, are computed by Ba (824); thus

wou, 107 or

Sd Coe aes a
0 mr |

a 0)

r= 00 | my = ag] CU) = St

Consider next the delta rosette, which is oriented at angles of 0%, 60, and 120°, as
shown in Fig. 3.13. With Py = 0°. 8), = 120°, and yy = 240°, Eq. (3.19) becomes

6 = 3 ent en) el (326)
2v3
Yo = em = en)

Therefore the principal strains and the principal angles can be expressed as

atan, la apo

à i
me am

{eu

a+ (e „tat
Vite el

3

DEC

o nm nl

also] eos)

oe

24. 35 Strain Gage Rosette Data Analysis and Correction 79

For convenicnes, a summary of the analytical expressions for four commonly
employed rosettes is presented in Table 3.1,

In the discussion above, itis assumed that any necessary corrections, such as
those for gage factor and for transverse sensitivity of the strain gages, have been made,
In Section 25, the general equation, Eg. (2.18), was derived for correcting transverse
effects in biaxial strain state. It can also be employed directly for the two-element rec»
tangular rosette. For other rosettes, however, new equations are necessary. In Example
3.12, the equations for correcting transverse Sensi a three-element rectangular
rosette will be derived,

EXAMPLE 3.12
Show that the equations for correcting transverse sensitivity in a Uhree-element rectangular
rosete, as shown in Fig, E3.12a,are given by

7 Oel - Kiel

ey QU + Kien = KA + eid]

eu = Olein ~ Kei}
i the ans:
ME oa ‘and is assumed the same fo ll thee element san
m jn

FIGURE F3.12 to o

cu = Dein
Imagine that there were another gage IV, mounted inthe direction perpendicular to gage Il, as
‘shown in Fi, E3.12b,Thus using Eq. (220).

er = Oley Ki)

an = Oley — Ket)
“According o the theory of elasticity, the sum of st
invariant, s0 we have

y two perpendicular directions is an

25. ones weet

RS sy edu PT DO ug Tay send jo Ga VE STEVE

26. 35, Strain Gage Rosette Data Analysis and Correction 81

i ak ie
ER
me
ee OE
= ;
a= 061 Ru
pare, af ere]
ie
| wii
5
Dota reste AN
|

and
teur eh + ey
Therefore,
= Qlein— Keto) = Oleh = Kite +

eu = Of + Ke = Kei + ei)

By the same procedures, ll equations in Table 32 can easily be obtnined. The key step is o.
‘obtain the stain invariant (e+ <3) for various rosetes.

EXAMPLE 3.13

‘Assume thatthe transverse sensitivity factor K, = 2.1% forthe rectangular rosette in Example
3.11, Calculate the principal strains and the principal suesses after correction of the transverse
sensitivity effects

27. 82 Chapter} — Strain Gage Circuitry, Transducers, and Data Analysis,

Solution By using expressions for the three element rectangular rosette in Table

obtains
2-
ee 0904(1 + DOI) — (ut) 4 12003] = ale
en = 0994/1200
ey eu = 1e
44> Sip

Den = 6 = fan = 21708) — 1558 = Mile

let [Esa o,
2 ET ar ens
quon 24) grt
u un
eB ps = ae
un BR a,
PER
Be 625)
De
E ae ah asie
pe Bt Noa <a

‘Similarly, equations for correcting transverse sensitivity for the delta rosette
can also be obtained. For completeness, a summary of expressions to correct trans»
verse effects for three commonly used rosettes is presented in Table 3.2. [1 is pointed
‘out that the transverse sensitivity factor is assumed to be the same for all elements
in Table 32. IF it is not true, the reader is referred to, for example, reference [5] for
details.

3.6 CORRECTION FOR WHEATSTONE BRIDGE NONLINEARITY

AS mentioned earlier, a nonlinearity effect must be considered when large strains (et.
S% of greater) are to be measured by using an unbalanced Wheatstone bridge. For a
lange class of commercial static strain indicators and signal conditioners employing an

28. 3.5. Correction for Wheatstone Bridge Nonlinearity 83

unbalanced Wheutstone bridge, it is necessary to provide a simple means for correcting,
the error due to the Wheatstone bridge nonlinearity (3.

Consider a general Wheatstone bridge, as shown in Fig. 3.14. Assume that gage R,
is subjected to a strain e. gage R; 10 a strain ae, gage R, to a strain be, und gage R (0 a
strain ce. In practice, a=0, 4, 07 -1;b =, y, or land e = 0, 1, or -1. Without loss of
generally, we assume that the bridge is initially balanced and R,= Ry= Ry= Ry =
Furhermor, eau tat ae pag Fst: et on the tla neta Cr sgn
conditioner): thus AR 4
bridge nonlincait I usually neplected in commercial srain indi
¿itioners and Eq, (3.4) is used to obtain the indicated strain €, which differs from the
actual strain e when large strains are involved, Since m= 1, Eq. (3.4) becomes

v
SE ¿(a rb- Ge

or

e ABE

bode=

VG, om

where (I a + h— che" is the indicated strain read directly from the strain indicator.
However, the actual bridge output is given by

+ AR) (Ry + AR;) — (Ri + ARR + AR

PAR, + Ra AR) CR) + AR, + Ry + AR)

aE
v

GH SRA + DARIA) = [1 + ARRIBA AN
ei = 0 +0 RATE = + DRA
„Urarb-gaRı a = 0 + D 0) + tb - ae) (ARYA
i Ret b js Ml tot E age]
Substituting Eq, 3.31) into Eg, (330) gives
Sl a+ be) + - aGele

e= ee — 65

ab Qi A var 1 JG + (I + a) + Ge

FIGURE 2.14 Epica Wheatstone
ide.

29. 34 Chapter3 Strain Gage Circuitry, Transducers, and Data Analysis

Equation (3.32) is a useful expression when corrections are necessary for Wheatstone
bridge nonlinearity. Since a, À and e are known beforchand in practice, Ihe corrected
strain € can he obtained by solving Eq. (3.32). À few cases used frequently in practice
are presented next,

Case 1: Quarter Bridge ( = 0). Only one active gage is used, such us in
Uniaxial tension or compression experiments This is also the case frequently used in
general two-dimensional strain measurements using strain gages. In this case,
Eq. plified as

4

rag

Solving the equation above for the corrected e gives

635

Case 2: Half Bridge (a = —11, b = € = 0). Two active gages are used. This is the
‘case in a uniaxial stress Held with one gage in the uniaxial stress direction and the other
in the transverse direction. The corrected strain « can be computed using the equation,

630

Ge

{

Case & Full Bridge ( 1 b= D. Four active gages in uniaxial stress field
are used, two gages are honded slong the uniaxial stress direction and two gages in
insverse direction, The corrected strain € is computed using the equation
2

rec 635

There are three special eases (o = 1, b = € = Qu = d = pe -lias ce -L.b= 1)
where the Wheatstone bridge output is always linear. In other words, no corrections
are necessary no matter how large the measured strains are.

EXAMPLE 3.14
Ti obtain a uniaxial stress-strain curve for a material up to large stain a room temperature, a
high-clongation strain gage was bonded Lo the specimen and a quarter bridge was used Ata

‘certain teile loa, the indicated strain was 140,000us: Determine the strain after correcting for
the bridge nonlinearity and the error introduced if a correction had not been made, The wage
factor set on the strain indicator was 2.1. If the load is compressive (the indicated tra is
—MOLOI e), what are the curreetel strain € and the error inroducod ifthe costo

made?

Solution Substiating €;

Mande = 1.14 into Bq,

gives

30. 3.7 Gage Factor for inite Deformation 85

pora
2- eno = 98
0165 ~ 014 so) =

error = SHEP (oo) = 152%

‘When compressive load was applied, the indicated st
ande =-0.18 into Eq (3.33) gives

live. Substtucing G, = 2.17

A)

018}
12
From the example above i is clear that the correction for the Wheatstone bridge nonlinearity is
very important when large stains are to be measured by strain gages

(100%) = 148%

3.7. GAGE FACTOR FOR FINITE DEFORMATION

‘As mentioned earlier, gage factor variation is another major error source in high-
clongation strain measurements by strain gages Tis a common practice to use 2 + € as
the gage factor correct in theory based on some simple assumptions bul not, to out
knowledge, substantiated by actual test results [4] However, wo found that the gage
{actor varies approximately according tothe equation below [9]

= G+ Ca 636)

where €, and C, are constants and e is the nominal strain, respectively. €, is usually
close to 2.0, but C; is usually greater than 1.1 and also close to 2.0. For example, C,
1.94 and C; = 1.82 for a KFE-2-C1 gage, obtained by a uniaxial compression test on an
‘annealed aluminum 1100 cylinder specimen and valid for strains larger than 0.04, as
shown in Fig. 3.15.

a

22] expesimomal data
yz 198-1806

ie |
“te
Nominal train (%)

FIGURE 3.15 Gage factor vacios with he ssi I

31. 86 Chapter3 Strain Gage Circuitry, Transducers, and Data Analysis

It is difficult to provide a unique value for both constants C, and C;, valid for
any strain gages up to large strains, The values determined by experiments may
depend on the gage materials, the lot number of the gages. the adhesive used to bond
the strain gages, test conditions (Vensile, compressive, or bending experiments}, speci
men material used. and specimen surface conditions (curved or fat) during the ealibr
tion,

3.8 SHUNT RESISTANCE CALIBRATION

Calibration of steain gage instrumentation either nstrum y or instrument
verificacion [7] is frequently done by indirect method, A resistance change (usually
decreasing the resistance of u bridge arm) is produced by shunting a large resistor
across one bridge arm to simulate a strain gage output. The basic shunt calibration of

For simplicity and without loss generality, we assume that X= R;= Ry= & and
R,= R, (quarter bridge). Thus the bridge is initially balanced, By shunting À, resis-
tance decrease in bridge arm 2 is produced, which is given by

+R
where R, is the calibration resistor. Since R, = R, and ARIR, = Ge, we have
Rs
E GAR +R) (338)

where e, is the strain simulated by shunting R, with R. Equation (3:38) can be used to
determine the value of calibration resistor for a given e. For example if ¢, = Sip
and Gy=2. R= 118900.

By using Eq. (2.4), it can be scen that a resistance deeresse in bridge arm 2 by
shunting. a calibration resistor across R, will produce a positive output: in other words,
à tensile strain is registered. For example. this calibration method is used in a P-3500
digital strain indicator |).

It has been discussed in Section 3.3 that when the gage is remote from the insiru-
ment, the lead wire resistance must be taken into account. The usual way to compen

FIGURE 2.16 sic sont elvan
Whearscone ie ce

32. 3.9 Potentiometer Circuit 87

sate for the error introduced by the lead wires isto adjust the gage factor. This can cas-
ily be done by shunt calibration. For example, ifthe measurement is to be done using a
P-3500 digital strain indicator and a quarter bridge is to be used, corrections for any
signal attenuation due to lead wire resistances can be done in the following steps:

1. Calculate e as follows:

x Oe

where Gis the gage factor provided by the manufacturer and SUVO e ds the
strain simulated by shunting R, across R, (= R,) when the gage factor is set at
2.00,
2. Depress the CAL button,
3. Adjust the gage factor control knob until the strain indicator registers ast
then lock the knob,
‘Shunt calibration is simple in principle, but actually is more complicated. The
reader is referred to reference [7] for more details.

3:9 POTENTIOMETER CIRCUIT
In dynamic strain gage applications, the potentiometer circuit is often useful, especially
when the circuit output consists of two parts; one Varies with time and the other is a
constant. A typical potentiometer circuit is shown in Fig. 3.17. The voltage drops across
R, or the voltage output of the potentiometer circuit, Vp i given hy the equation,

R y

ee 639
where » and V is the input voltage.
‘Consider the case that resistances Rand R are changed by the amounts AR, and

AR, The voltage output of the potentiometer cite
namely,

can be computed using Eq. (3.39),

Rt AR,

MN er ETA

FIGURE 3.17. Topic potentiometer

33. 88 Chapter3 Strain Gage Circuitry, Transducers, and Data Analysis

Ifthe nonlinearity of the potentiometer circuit is neglected (e. in the ease when the
strain 10 be measured is Less than 1%), the output inerement, AV.,,, can he expressed

bythe equation
mv
Sat ara

640)

is the basic equation governing the strain measurement us
circuit

“The sensitivity of potentiometer circuit, S is defined by
ESA (2 on

TE ANT ie
Since Egs. (3.40) and (3.41) are exactly the same as those of an intially balanced half
Wheatstone bridge, some previous discussions regarding quarter and half bridges, such
as temperature compensation, sensitivity, lead wire effect, and nonlinearity. are equally
valid for the potentiometer cireuit. However. the actual output of a potentiometer cir
cuitis Vag + Wy

In general, AV 4g is much smaller than Vy in magnitude. Thus it cannot be mes

sured accurately when both AV 43 and V 4, are constant with respect to time; this is the
«ase in the static train measurement, However, in dynamic train applications, AV yy is
ng with time but Y, is constant with respect to time. The constant component
fof the potentiometer circuit output, Vs, can be blocked by using filters, In this way
AV scan be measured accurately.

potentiometer

Ss

PROBLEMS

31. Verify Ey. (3.3) ifthe second-order terms are neglected

32. To increase the circuit sonsitivit the voltage can he varied in the quarter bridge, shown
in Fig, 132 The active gage isa 350-0 gage dissipating 0.05-W power with a gage factor
of 2.12. Caleulate the required voltage to be applied to the bridge and the resulting cir
cuit sensitivity

Ti

FIGURE P32

33, Prove thatthe bridge arranges

illustrated in Fig 35 is temperature compensated.

34. Problems 89

3A. The circuit semis forthe bridge arrangement,
‘What could the theoretical value of maxinium seas
Kenn

35, Prove thatthe bridge arrangement shown in Fig 37 is temperature compensated i all

es experience the same temperature change.

36. In addition toa bending moment M a tensile load Ps also applied on the beam shown
in Fig. 374. Show that the output AE is independent of the load P if full bridge

(Fig. 879) is employed. Assome that Ihe era section are of the beam is À

To eliminate possible bending effect due to coventrc loading to climinate effet due to

temperature change, and to increase the circuit sensitivity, four gages are mounted un a

tensile spocimon, as shown in Fig P37. Determine the positions of the four sages in the

bridge and derive an expression to relate he bridge output AA with the axial strain , (a)

ia lf bridge is o be used, and (b) if ful bridge isemployed. Assume that a fixed volt-

age V is applied to the bridge. The gage resistance, Poisson" ratio, and gage factor ate R,

sand G respectively

33. given by Eq. (3.10).
be ifthe voltage were variable?

30.

namen te

38. In Example 34, how should the four gages be placed in the bridge if a full Wheatstone
bridge isto be used? What is Ihe output AE in tems of Y Gp. ande?

I quartercircuae gages (refer to Problem 2,7) ae located in adjacent quadrant, as

shown in Fig 39. A half Wheatstonc bridge is employed. Show that the bridge output

ABIV is proportional tay, =. (Transverse sensitivity elects are neplected,)

FIGURE P59

35. 90 Chapter3 Strain Gage Circuitry, Transducers, and Data Analysis

40, Four quarter<ircular gages are located in al four q
Wheatstone bridge is used
a) Position the tour gages in the full bridge proper
proportional to shear stray,
(0) Determine the bridge output SEN

aan as shown in Fig, P3.10.A full

order to have the bridge output

(Transverse seasitvty effects are neglected)

FIGURE F3.10

3:11. Two two-slement shear gages (Fig. 23m rn) are mounted on a cylindrical piece at di
wotrially opposite locations a shown in Fig. PL.

(a) Position the four elements in à half Wheatstone bridge in onder (0 have the bridge
‘output proportional to shesr strain {ur torque 7) and independent of bending
moment M and axial Jose Pe

(@) Determine the bridge output AEIV and prove that it is independent of M and R
‘The cylindrical piece has circular eros section with radius of R and à shear moda
us of 6

\ Y

Faure Past

12. A cantilever beam with four strain gages is wei asa load transducer, as shown in Fi
PRAD.A full Wheatsione bridge i o he used, Show the postions of four pages in the

36. Problems 91

bridge so thst maximum sensitivity is obtained and the output is equal to (VG -dhi2 EN)
“ment oa the dimension din relation o sensitivity: Note that Vand Gare the bridge
xcituion voltage and gage factor. respectively.

e
Pa
Mw x N
A a is,
er Da
FIGURE 93.12. El 2

MA. An extenson

+. shown in Fig 3.10, consists of à semicieular-arch clip gage. The
EL

iE

“applied on the clip gage Place gages Land 2 a Wheatstone bridge and determine ALI

dimension ofthe clip gage is shown in Fig. PA. where AL = pa Pis tie force

approximately the length ofthe clip ge.

in terms of €, G 6 and Ken e

FIGURE P2213

3.14, Determine AF of an aoselerometer shown in Fig. MIC. The dimensions of the beam are
shown in Fig.P3.14, Young's modulus ofthe beam is E

FIGURE F3.14

3.18, Using Mohr‘ train cece, prove thatthe principal ample ey forthe rectangular rosette
shown in Fig 312,38

37. 92 Chapter3 Strain Gage Circuitry, transducers, and Data Analysis

PER ru en
SS ee ker

Ber Meng ad ep
a, = 490" whene < eq and ey = Lat

M6. A three-element rectangular rosette is mounted on & steel pressure vessel at the region
ining the spherical head and the clindrical shell (Fig. P3.6). At a pressure of SM pi,
the strains indicated for elements 4, 5, and 6 are, respectively, due, Oe, and 1236
The gage factor is 2.8, but the value set on the strain indicator is 2.12, Determine the
principal stresses and strains and the principal angle. Young's modulus for ste is 0.000,

li Poisson's ratio i (1290.

(eer

Zen)

newer N.

IT. A delta rosette is mounted on a spherical head of a steel pressure vessel as shown in Fig
PLAT. At a pressure of 500 psi, the stains indicated for the three gages 1.2, and 3 re.
respectively, ó4pe, Te, and te, Determine the principal stresses and stains and the
principal angle. The gaye factor 2.12, which is set on the strain ndientor, Young s mod-
‘ulus and Poisson's ratio for steel are 36,000 ksi und 1.200

|)
Ss
nourepass ON AY

3:18, Determine expressions for principal strains and stresses for a delta rosette shown in
acy of

Fig. 3.18, Note thatthe advantage of adding a fourth gape is 1 check on thea

AN

m
Fer

a
FIGURE 3.98 ly

249, Prose that the equation for correcting the transverse sensitivity for len
Teta rosette, shown in Fig, 13.18, given by

en = Ole Kiel + in — ei

38. References 93

she Qu (Ln G= Land IV) are indicated strain and ey i corrected
Strain, respsctvely Assume that all elements have the same transverse semitvity Ec
work,

320. Show that the expression for correcting transverse strain effects or element II in a delta
rosete (listed in Table

e Jaca + o]

‘Assume that elements LI and IIT have the same transverse sensitivity factor X. Q =
GR — Ke) (@= 1, and II) are indicated strains, and e 8 the corrected
strain for element IL

321. In Problem 3.16, the factor of transverse sensitivity forthe theee-clement rectangular
rosette is +2.1%. Determine the principal stesses and the principal angle. Alo deter-
mine the error in determining the maximum principal stresses ifthe effet of transverse
sensitivity is neglected

322. In Problem 3.17,the delta rosette has a factor of transverse sensitivity 42.3%, Determine
the principal stresses and the principal angles. How much error will be introduced in
‘determining the maximum principal sires i the effect of transverse strains is neglected?

323. Repeat Problem 321,

324, Repeat Problem 32218 K,

REFERENCES

[1] €.Wnentstune, An account of several new instruments and processes for determining the
«onstats of voltaic cru, Pas Frans R. Soc (London), 33, 3031843.

121 WM. Murray and PK Sci, Sri Gage Techniques 1958.

131 Wheaatone Bridge Nontóncoi Tech. Nose TN-S07, Measurements Group, Inc. Raleigh
(North Carolina) 1982

14) Hiph-Eiongaion Measurements, Tech Tip TTS, Measurements Group, Inc, Raleigh
(orth Carina), 108.

15) Transverse Sensitivity Erors Tech. Note TN
(ont Carlin), 1982.

{61 A.S.Khan and M. Maik, Elementary Solid Mechanics in review

IM Shunt Calibration, Tech. Note TNSi4, Measurements Group, Inc, Raleigh (North
Carolina), 1988

(81 P3500 Diga! Stain Indiotor Inte. Manual, Measurements Group, Inc, Raleigh
(North Carolina). 198.

191 S Huang and A. Khan, On the une of elecrza-resitance metalic oi stain gages or
measuring Jure dynanie plastic deformation up to fony percent, Exp, Mech. vol 31,

, Measurements Group. Toc. Raleigh